For the other values of \(q\)  (i.e. \(q > 243\) ) we have not been able to explicitly evaluate \(H(\Gamma )\)  since it seems that there are too many variables for this problem to be treated even with the help of a computer. Hence we do not provide an explicit upper bound on \(EGZ(3,\mathbb {F}_{q}^{n},2)\)  for \(q>243\) . Also developing the product that defines \(P\)  and trying to group the slices as done by Naslund and Sawin in [1]} appears not helpful in this problem.
On the other hand, we believe it is interesting that we can prove, just theoretically, the upper bound of Theorem REF .
